In the diagram, $AB = 25 \mbox{ cm},$ $AC = 20 \mbox{ cm},$ and $\angle A = 90^\circ.$ What is the area of triangle $ABC?$

[asy]
draw((0,0)--(25,0)--(0,20)--cycle,black+linewidth(1));
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black+linewidth(1));
label("$A$",(0,0),SW);
label("$B$",(25,0),SE);
label("$C$",(0,20),NW);
[/asy]
Explanation: Triangle $ABC$ has base $AB$ of length $25\text{ cm}$ and height $AC$ of length $20\text{ cm}.$ Therefore, the area of triangle $ABC$ is \begin{align*}
\frac{1}{2}bh &= \frac{1}{2}(25 \mbox{ cm})(20 \mbox{ cm}) \\
&= \frac{1}{2}(500 \mbox{ cm}^2) \\
&= \boxed{250} \mbox{ cm}^2.
\end{align*}